Two dice are tossed. What is the probability that the sum of both dice is a prime number?
this is the answer i have ;
Presuming two fair, 6-sided dice, each numbered 1 through 6:
You can achieve one of the following results on any given roll: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. But there is only 1 way to get a 2, 2 ways to get a 3, 3 ways to get a 4, up to 6 ways to get a 7, then 5 ways to get 8, and so on. Total is 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36 possible outcomes.
The prime numbers from 2 through 12 inclusive are: 2, 3, 5, 7, and 11. There is 1 way to get a 2, 2 ways for a 3, 4 ways for a 5, 6 ways for a 7 and 2 ways for 11. Total of 1 + 2 + 4 + 6 + 2 = 15 successful outcomes out of 36 possible outcomes.
15/36 = 5/12 = 0.417
Right.
To find the probability that the sum of two dice is a prime number, we need to determine the number of successful outcomes and the total number of possible outcomes.
First, let's analyze the possible outcomes. When two dice are tossed, each die has 6 possible outcomes since they are fair and have numbers 1 through 6. Therefore, the total number of possible outcomes is 6 * 6 = 36.
Next, we need to determine the number of successful outcomes, which are the outcomes where the sum of the two dice results in a prime number. The prime numbers between 2 and 12 (inclusive, since that is the total sum range) are 2, 3, 5, 7, and 11.
To calculate the number of successful outcomes, we need to count the number of ways we can get each of these prime numbers as the sum. We can achieve a sum of 2 in only one way: rolling a 1 on both dice. There are two ways to get a sum of 3: rolling a 1 and 2 on different dice or rolling a 2 and 1 on different dice. For a sum of 5, we have four possibilities: 1 and 4, 4 and 1, 2 and 3, or 3 and 2. The sum of 7 can be achieved in six ways: 1 and 6, 6 and 1, 2 and 5, 5 and 2, 3 and 4, or 4 and 3. Finally, the sum of 11 has two possibilities: 5 and 6, or 6 and 5.
Adding up the successful outcomes, we get 1 + 2 + 4 + 6 + 2 = 15.
So, the probability of getting a sum of both dice as a prime number is 15 successful outcomes out of 36 possible outcomes. This can be written as 15/36, which can be further simplified to 5/12 or approximately 0.417.